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Grade 2 · 1.NBT.C.6 · 2.NBT.B.5

Subtracting two-digit numbers without regrouping

The bookmobile has 87 books on board and children borrow 34 of them. Counting back thirty-four times would take a long time. Is there a shorter way to see what is left, and why does it give the right answer?

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Subtracting within tenKindergartenTens and onesGrade 1
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Regrouping in subtractionGrade 2
Where you are 7 in Multi-digit subtraction
Number9
Arithmetic5
Addition and subtraction12
Multi-digit subtraction7
Subtracting multiples of ten
Subtracting two-digit numbers without regrouping
Regrouping in subtractionSubtracting three-digit numbers by place value, including across zeroThe standard subtraction algorithm▾ Show all 7 (2 more)

When you reach for it

A child who knows Subtracting within ten can find 7 - 4 at once. Now the numbers are bigger. The bookmobile has 87 books and children borrow 34. Counting back thirty-four steps from 87 works, but it is slow and one slip spoils it. There is a shorter way, and it rests on Tens and ones: a two-digit number is some tens and some ones.

87 is 8 tens and 7 ones. 34 is 3 tens and 4 ones. Taking away 34 means taking away 3 tens and 4 ones. Tens come from the tens and ones come from the ones:

  • 8 tens take away 3 tens leaves 5 tens;
  • 7 ones take away 4 ones leaves 3 ones.

So 5 tens and 3 ones are left, and 87 - 34 = 53. The two small facts, 8 - 3 and 7 - 4, are ones the child already knows.

This lesson covers the case where each place has enough: the first number has at least as many tens as the second, and at least as many ones. In 87 - 34 there are 8 tens to take 3 from and 7 ones to take 4 from, so nothing has to be broken apart. When a place does not have enough, as in 82 - 34 (only 2 ones, and 4 to take), a ten has to be opened first. That is a different move, named under Boundaries.

It is the subtracting twin of Adding two-digit numbers without regrouping: the same rule, tens with tens and ones with ones, run the other way. The pictures are ones the child knows: bundles of ten sticks with loose sticks, base-ten rods and units, a quick drawing of the same thing, and an open number line.

The procedure

The three days build one idea. Day 1 takes away with rods and units, then with a drawing. Day 2 reads the same move in the digits and uses it to find how far apart two numbers are. Day 3 checks an answer by adding back, walks the subtraction on an open number line, and finds a missing number.

Day 1: take tens from tens and ones from ones

Build the first number. Ask the child to show 87 with base-ten pieces: 8 rods of ten and 7 units (loose ones). Bundles of ten sticks do the same job. A rod is counted as one ten without opening it, which is the picture from Tens and ones.

Say what is being taken. The number taken away, 34, is also tens and ones: 3 tens and 4 ones. Say it out loud before touching anything. “Thirty-four is three tens and four ones.”

Take each kind from its own kind. Take 3 rods away from the rods. Take 4 units away from the units. Rods are only ever taken from rods, because a rod is a ten and the 3 tens of 34 are tens. What is left is 5 rods and 3 units, which is fifty-three.

5 tens left 3 ones left
Figure 187 - 34. Three tens are crossed out of the tens and four ones out of the ones. Five tens and three ones are left.

Nothing else happened. The 5 rods left over were never touched by the ones, and the 3 units left over were never touched by the tens. That is why the two small subtractions can be done separately and then put together: the tens and the ones are two separate piles, and 34 takes some from each. This is taking away, the first meaning of subtraction from Subtraction as taking away.

Which place first? Here it does not matter. Taking the rods first and the units second, or the other way around, leaves the same pieces, because neither pile is short. Many children take the tens first because that is how the number is read, left to right. The standard written way works from the ones first, for a reason that only shows up when a place runs short; that is The standard subtraction algorithm.

Draw it. Once the pieces are familiar, a quick drawing does the same job faster. A tall line stands for a ten and a dot for a one. To find 48 - 16, draw 4 lines and 8 dots, then cross out 1 line and 6 dots. Count what is not crossed out: 3 lines and 2 dots, so 32. The drawing is the rods and units on paper, and crossing out is taking away. Example 2 does this.

Day 2: the digits, and finding the difference

The digits tell the story. Write the two numbers as tens and ones, one under the other:

tensones
8787
take away 3434
left53

Each column is a subtraction the child already knows. The tens column says 8 - 3 = 5, which means 80 - 30 = 50, as in Subtracting multiples of ten. The ones column says 7 - 4 = 3. The answer’s tens digit comes from the tens column and its ones digit from the ones column. This works because 87 = 80 + 7 and 34 = 30 + 4, and

87 - 34 = (80 - 30) + (7 - 4) = 50 + 3 = 53

The step that splits the subtraction into two is allowed only because each column has enough. That is the whole condition for this lesson.

A place may end at zero. If the two tens digits are the same, no tens are left, as in 67 - 63: the tens column is 6 - 6 = 0 and the ones column is 7 - 3 = 4, so the answer is 0 tens and 4 ones, written 4. If the ones digits are the same, the answer ends in 0, as in 53 - 23 = 30. Example 4 does both with other numbers.

Finding the difference. Subtraction also answers “how many more?” and “how far apart?”. The bookmobile carries 98 picture books and 46 chapter books. How many more picture books are there? Nothing is taken away here; the two groups are compared. Lay out 98 in one row and 46 under it, and match each rod to a rod and each unit to a unit (Figure 2). The pieces of the top row with no partner are the difference.

98 46
Figure 298 matched against 46. The shaded pieces have no partner. They are the difference, 5 tens and 2 ones.

The unmatched pieces are 5 rods and 2 units, so there are 52 more picture books. That is the same as taking 46 from 98: 9 - 4 = 5 tens and 8 - 6 = 2 ones. Matching tens with tens and ones with ones is exactly the place-value move again. This is the comparing meaning of subtraction from Subtraction as comparison, and it is why a story with no taking away in it can still be a subtraction.

Order matters. 98 - 46 asks how much more 98 is than 46. Turned around, 46 - 98 asks to take 9 tens from 4 tens, and there are not enough. With whole numbers (zero and the counting numbers) there is no answer. In “the difference between 46 and 98”, the larger number goes first.

Day 3: check by adding back, jump back, find what is missing

Adding back checks the answer. Subtraction undoes addition. If 34 was taken from 87 and 53 was left, then putting the 34 back should give 87 again:

53 + 34 = 87

This is the tens with tens, ones with ones addition: 5 + 3 = 8 tens and 3 + 4 = 7 ones. When the subtraction had enough in each place, adding back makes no new ten, because each column just returns to what it was. If the sum does not come back to the first number, something went wrong, and the child knows to look again.

Jumping back on an open number line. An open number line has no scale marked; the child marks only the numbers used. To find 79 - 36, start at 79. The 36 is 3 tens and 6 ones, so jump back by ten three times: 69, 59, 49. Then jump back 6 more, to 43 (Figure 3). A child sure of the tens can make one long jump of 30 and one of 6. The jumps of ten keep the ones digit at 9; only the last jump changes the ones.

A missing number. 78 - \square = 32 asks how much was taken. The tens went from 7 to 3, so 4 tens were taken. The ones went from 8 to 2, so 6 ones were taken. The box is 46. Adding back checks it: 32 + 46 = 78. Thinking of a subtraction as an addition with a part missing is Subtraction as an unknown addend, and counting up from the smaller number is Counting up to find a difference.

Worked examples

Read each problem aloud. Base-ten pieces or bundles of sticks on the table let the child build every one. When they can say the answer by thinking “tens from tens, ones from ones”, they are subtracting by place value in their head.

Example 1

Day 1: Books borrowed from the bookmobile

The bookmobile parks outside Waru’s school with 87 books on its shelves. During the visit, children borrow 34 books. How many books are still on the shelves?

Solution

Books leave the shelves, so we take away: 87 - 34.

87 is 8 tens and 7 ones. The 34 books borrowed are 3 tens and 4 ones. Tens come from the tens: 8 - 3 = 5 tens. Ones come from the ones: 7 - 4 = 3 ones. Each place had enough, so nothing had to be opened. That leaves 5 tens and 3 ones:

87 - 34 = 53

There are 53 books still on the shelves. Figure 1 shows the same subtraction with the borrowed pieces crossed out.

Check by adding back: 53 + 34 is 5 + 3 = 8 tens and 3 + 4 = 7 ones, which is 87, the number the bookmobile started with.

Example 2

Day 1: Lemons picked from the tree

The lemon tree in Chimamanda’s yard has 69 lemons on it. Chimamanda picks 25 of them. Use a quick drawing to find how many lemons are left on the tree.

Solution

Lemons come off the tree, so we take away: 69 - 25.

Draw 69: 6 tall lines for the tens and 9 dots for the ones. The 25 lemons picked are 2 tens and 5 ones, so cross out 2 lines and 5 dots. Lines are crossed out only among the lines and dots only among the dots, because tens are taken from tens and ones from ones.

Not crossed out: 4 lines, since 6 - 2 = 4, and 4 dots, since 9 - 5 = 4. That is 4 tens and 4 ones:

69 - 25 = 44

There are 44 lemons left on the tree. The two digits of the answer are the same, but they mean different things: the first 4 counts tens (forty lemons) and the second counts ones (four lemons).

Example 3

Day 2: How many more picture books?

The bookmobile carries 98 picture books and 46 chapter books. How many more picture books are there than chapter books?

Solution

Nothing is taken away. The question compares two groups, so we find the difference: 98 - 46.

Match the two groups piece by piece, as in Figure 2. Each of the 4 tens of 46 matches one of the 9 tens of 98, and 9 - 4 = 5 tens have no partner. Each of the 6 ones of 46 matches one of the 8 ones of 98, and 8 - 6 = 2 ones have no partner. In tens and ones:

tensones
picture books98
chapter books46
how many more52
98 - 46 = 52

There are 52 more picture books than chapter books. Check by adding back: 52 + 46 is 9 tens and 8 ones, 98. In words, 46 chapter books and 52 more make as many as the picture books.

Example 4

Day 2: When a place ends at zero

Find 64 - 24 and 58 - 53. Then say what the zeros mean.

Solution

For 64 - 24: 6 tens take away 2 tens leaves 4 tens. 4 ones take away 4 ones leaves 0 ones. That is 4 tens and no ones:

64 - 24 = 40

The 0 must be written. It holds the ones place so that the 4 stays in the tens place; without it the answer would read 4, which is four ones, not forty.

For 58 - 53: 5 tens take away 5 tens leaves 0 tens. 8 ones take away 3 ones leaves 5 ones. That is 0 tens and 5 ones, which is just five:

58 - 53 = 5

Here the zero is not written. A number does not start with a 0 digit: 0 tens and 5 ones is written 5, not 05. Both answers are right, and both checks agree: 40 + 24 = 64 and 5 + 53 = 58.

Example 5

Day 3: Lemons sold, checked by adding back

Nalu and Waru pick 76 lemons from the lemon tree. They sell 43 of them to neighbors. How many lemons do they have left? Check the answer by adding back.

Solution

Lemons are sold and leave, so we take away: 76 - 43.

76 is 7 tens and 6 ones; 43 is 4 tens and 3 ones. Tens: 7 - 4 = 3. Ones: 6 - 3 = 3. Both places had enough.

76 - 43 = 33

They have 33 lemons left.

To check, put the sold lemons back with the ones left: 33 + 43. Tens with tens, 3 + 4 = 7; ones with ones, 3 + 3 = 6. That is 76, the number they picked:

33 + 43 = 76

The check works because adding 43 undoes taking away 43. If it had come back as any other number, the subtraction would need another look.

Example 6

Day 3: Bookmarks on an open number line

The bookmobile brings 79 bookmarks and gives away 36 of them. Show on an open number line how many are left.

Solution

Bookmarks are given away, so we take away: 79 - 36. The 36 is 3 tens and 6 ones.

Mark 79 on an open number line. Take away the tens first, as three jumps back of ten each:

79 \to 69 \to 59 \to 49

Each jump of ten changes the tens digit and leaves the ones digit at 9. Now take away the 6 ones as one short jump back, from 49 to 43. Only the ones digit changes, 9 - 6 = 3.

43 49 59 69 79 −10 −10 −10 −6
Figure 379 - 36 on an open number line. Three jumps of ten back, then one jump of six.

There are 43 bookmarks left: 79 - 36 = 43. The jumps show the place-value way in motion: three tens off the tens, then six ones off the ones. The same answer comes from the digits, 7 - 3 = 4 tens and 9 - 6 = 3 ones.

Example 7

Day 3: How many books were borrowed?

The bookmobile leaves in the morning with 96 books. At the end of the day it has 51 books. How many books were borrowed?

Solution

The books borrowed were taken from the 96, and 51 were left. So 96 - \square = 51, where the box is the number borrowed.

Look at each place. The tens went from 9 to 5. Since 9 - 4 = 5, 4 tens were taken. The ones went from 6 to 1. Since 6 - 5 = 1, 5 ones were taken. So 4 tens and 5 ones were taken:

96 - 45 = 51

45 books were borrowed. Check by adding back the borrowed books: 51 + 45 is 5 + 4 = 9 tens and 1 + 5 = 6 ones, 96. The same box could be found by subtracting the other way, 96 - 51: 9 - 5 = 4 tens and 6 - 1 = 5 ones, again 45.

Boundaries

  • Each place must have enough. The ones digit of the first number must be at least the ones digit of the second, and the same for the tens. In 82 - 34 there are 2 ones and 4 to take, so a ten must be opened into ten ones first; that is Regrouping in subtraction.
  • The first number must not be the smaller. It may equal the second (55 - 55 = 0), but with whole numbers 34 - 87 has no answer: there are not 8 tens to take from 3. In a “difference between” question the larger number goes first.
  • Order matters. 87 - 34 and 34 - 87 are not the same, unlike addition, where the order of the two numbers does not change the total.
  • A zero in the answer. A tens place that ends at 0 is not written (58 - 53 = 5). A ones place that ends at 0 is written (64 - 24 = 40).
  • Two places only. The numbers here are below 100. With hundreds the same rule works place by place, in Subtracting within 1,000.
  • The second number may be one digit. 47 - 5 is 4 tens and 7 ones take away 0 tens and 5 ones, so it fits the same rule: 42.

Where this goes next

Taking tens from tens and ones from ones is the first time a child subtracts by place value, and every written way of subtracting larger numbers rests on it. The case named under Boundaries, where a place runs short and a ten is opened into ten ones, is the next step: Regrouping in subtraction. Writing the columns one under the other and working from the ones, with that opening built in, is The standard subtraction algorithm, and the same rule with hundreds is Subtracting within 1,000. Counting up from the smaller number, a different road to the same difference, is Counting up to find a difference.

Exercises

Day 1: take tens from tens and ones from ones

  1. 58 - 23 = \square
  2. 76 - 52 = \square
  3. 49 - 37 = \square
  4. 93 - 61 = \square
Figure 4The base-ten picture for Exercise 5. The crossed-out pieces are taken away.
  1. The picture shows a subtraction. The crossed-out pieces are taken away. Write the subtraction and find what is left.
  2. The lemon tree has 78 lemons. Waru picks 45 of them. How many lemons are still on the tree?
  3. A shelf in the bookmobile holds 67 books. Children borrow 32 of them. How many books are left on the shelf?

Day 2: the digits, and finding the difference

  1. 86 - 46 = \square
  2. 39 - 34 = \square
  3. 55 - 55 = \square
  4. Nalu counts 68 lemons on the tree and 23 lemons on the ground. How many more lemons are on the tree than on the ground?
  5. At the bookmobile, Waru reads 74 pages and Chimamanda reads 51 pages. How many more pages does Waru read?
  6. Find 68 - 42 by writing each number as tens and ones.
  7. Find the difference between 31 and 84.

Day 3: check by adding back, jump back, find what is missing

  1. Find 67 - 25. Then check your answer by adding back.
  2. Use adding back to decide whether 79 - 44 = 25 is true. If it is not, find the right difference.
  3. Show 95 - 32 on an open number line, jumping back by tens and then by ones. Which numbers do you land on?
  4. 47 - \square = 21
  5. \square - 14 = 52
  6. There were 89 lemons on the lemon tree. After some were picked, 57 are left. How many lemons were picked?

Explain

  1. In 75 - 32, why can you take the tens from the tens and the ones from the ones, and then put the two answers together?
  2. Which of these can be found by taking tens from tens and ones from ones, with no regrouping: 64 - 31, 64 - 38, 46 - 31? Find the ones that can.
  3. Why does adding back check a subtraction?
  4. Is 72 - 51 the same as 51 - 72? Explain.
  5. Find 85 - 4 and 85 - 40. Why are the answers so far apart?
Answers
  1. 35. 5 - 2 = 3 tens and 8 - 3 = 5 ones: 58 - 23 = 35.
  2. 24. 7 - 5 = 2 tens and 6 - 2 = 4 ones: 76 - 52 = 24.
  3. 12. 4 - 3 = 1 ten and 9 - 7 = 2 ones: 49 - 37 = 12.
  4. 32. 9 - 6 = 3 tens and 3 - 1 = 2 ones: 93 - 61 = 32.
  5. 57 - 23 = 34. The picture shows 5 rods and 7 units, which is 57. Crossed out are 2 rods and 3 units, which is 23. Left are 3 rods and 4 units.
  6. 33 lemons. 7 - 4 = 3 tens and 8 - 5 = 3 ones: 78 - 45 = 33.
  7. 35 books. 6 - 3 = 3 tens and 7 - 2 = 5 ones: 67 - 32 = 35.
  8. 40. 8 - 4 = 4 tens and 6 - 6 = 0 ones, and the 0 is written: 86 - 46 = 40.
  9. 5. 3 - 3 = 0 tens and 9 - 4 = 5 ones; 0 tens and 5 ones is written 5: 39 - 34 = 5.
  10. 0. 5 - 5 = 0 tens and 5 - 5 = 0 ones, so nothing is left: 55 - 55 = 0.
  11. 45 more lemons. Compare: 6 - 2 = 4 tens and 8 - 3 = 5 ones, so 68 - 23 = 45.
  12. 23 more pages. 7 - 5 = 2 tens and 4 - 1 = 3 ones: 74 - 51 = 23.
  13. 26. 68 is 6 tens and 8 ones; 42 is 4 tens and 2 ones. 6 - 4 = 2 tens and 8 - 2 = 6 ones, so 68 - 42 = 26.
  14. 53. The larger number goes first: 84 - 31. 8 - 3 = 5 tens and 4 - 1 = 3 ones, so 84 - 31 = 53.
  15. 42. 6 - 2 = 4 tens and 7 - 5 = 2 ones, so 67 - 25 = 42. Check: 42 + 25 = 67.
  16. Not true. Adding back gives 25 + 44 = 69, not 79. The right difference is 35: 7 - 4 = 3 tens and 9 - 4 = 5 ones, so 79 - 44 = 35, and 35 + 44 = 79.
  17. 85, 75, 65, then 63. Three jumps of ten back from 95 take away the 3 tens, and one jump of 2 takes away the ones: 95 - 32 = 63.
  18. 26. The tens went from 4 to 2, so 2 tens were taken; the ones went from 7 to 1, so 6 ones were taken: 47 - 26 = 21.
  19. 66. Adding back the 14 that was taken gives the start: 52 + 14 = 66, and 66 - 14 = 52.
  20. 32 lemons. 89 - \square = 57: the tens went from 8 to 5, so 3 tens; the ones went from 9 to 7, so 2 ones. 89 - 32 = 57, and 57 + 32 = 89.
  21. 75 is 7 tens and 5 ones, and 32 is 3 tens and 2 ones. Taking 32 away means taking 3 tens and 2 ones. The tens and the ones are separate piles, and each has enough, so 3 tens come off the 7 tens and 2 ones come off the 5 ones without either pile touching the other. What is left is 4 tens and 3 ones: 75 - 32 = 43.
  22. 64 - 31 and 46 - 31. 64 - 31 = 33 and 46 - 31 = 15. In 64 - 38 there are only 4 ones and 8 ones to take, so a ten must be opened first; that is regrouping.
  23. Taking a number away and then putting the same number back returns to where you began. So if the difference is right, adding the number taken away to the difference gives the first number. If it gives anything else, the difference is wrong.
  24. No. 72 - 51 = 21: 7 - 5 = 2 tens and 2 - 1 = 1 one. But 51 - 72 asks to take 7 tens from 5 tens, and there are not enough, so with whole numbers it has no answer. In subtraction the order matters.
  25. 81 and 45. In 85 - 4 the 4 is 4 ones: 5 - 4 = 1 one, and the 8 tens stay, so 85 - 4 = 81. In 85 - 40 the 40 is 4 tens: 8 - 4 = 4 tens, and the 5 ones stay, so 85 - 40 = 45. Four tens is forty, far more than four ones.
kg1128 · updated 2026-10-21 · CC-BY-SA-4.0